Mahler’s first conjecture (symmetric case)

Prove that for every origin-symmetric convex body K ⊂ R^n, the Mahler volume M(K) = n! |K| |K°| satisfies M(K) ≥ 4^n, with equality attained by both the Euclidean ball B_2^n and the cube B = [-1,1]^n.

Background

For a convex body K ⊂ Rn, the Mahler volume is M(K) = n! |K| |K°|, which is invariant under GL(n,R). Bourgain–Milman established a bound of the form M(K) ≥ cn for some universal c > 0, but the sharp constant is unknown.

Mahler’s first conjecture predicts the optimal constant c = 4 in the symmetric case, with extremals given by the Euclidean ball and the cube.

References

Mahler's first conjecture asserts that c should be 4 if K is symmetric, attained by both Bn and B [70, p. 96].

— Convex meets complex  (2410.23500 - Rubinstein, 2024) in Section 8

The symmetric Mahler conjecture (see ) suggests that the stronger inequality $$m(K)m(K\ast)\geq \frac{4n}{n!},$$ should hold.

— Helson Inequality, Hankel Operators, and Weak Factorization on Paley-Wiener spaces of Convex Domains  (2609.10730 - Bampouras, 9 Sep 2026) in Section 2, “Helson's Inequality”

It was shown in that Mahler's conjecture is equivalent to a conjecture about volumes of symplectically self-polar convex bodies. The latter conjecture states that

\vol X \geq \frac{2n}{n!}

for any symplectically self-polar convex body $X \subset \mathbb{R}{2n}$.

— Contractibility of space of symplectically self-polar convex bodies  (2608.13280 - Berezovik, 13 Aug 2026) in Section 1, Introduction

Mahler's conjecture , suggesting that the minimum of the product of the volumes of a centrally symmetric convex body and its polar is attained at the cube, has been proved in and for dimensions $2$ and $3$, but is still open for all dimensions $d>3$.

— Galerkin approximations to the space of convex bodies by polytopes in nondegenerate V-representation  (2608.26615 - Rieger, 27 Aug 2026) in Section 1, Introduction